This reminds me of a wonderful mathoverflow answer about desirable properties of math notation, authored by none other than Terrance Tao:
https://mathoverflow.net/questions/366070/what-are-the-benef...
https://mathoverflow.net/questions/366070/what-are-the-benef...
But since mathematicians absolutely love finding common patterns in things, maybe there's some new innovation that a representation with uniqueness would enable?
But we don't always want to work with normal forms, for one reason or another, and there can be multiple kinds of normal form to choose from depending on your needs. For instance, if you do anything with something in a normal form, chances are it's no longer in a normal form! The lack of closure properties like this means you may only normalize at the very end of a series of manipulations, during which you're using a more suitable notation.
I think this is important and applies well to programming languages. Many programming techniques and styles can be extremely powerful when continuously working in the area whereas just jumping in they can be totally maddening. Sometimes this sort of thing builds up a cognitive wall to newcomers. It has its uses, but it's frustrating when you're one of those casual users having to deal with the clever ideas in the codebase.
This is never the case in physics because the equality sign "=" is loaded and can mean equality, definition, proportionality, identity and equivalence. Physicists read the equality sign case by case, as the case may be, with one of these meanings. In scholasticism this used to be called casuistry.